# 1 22 polytope

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In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices).

## Contents

Its Coxeter symbol is 122, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 122, construcated by positions points on the elements of 122. The rectified 122 is constructed by points at the mid-edges of the 122. The birectified 122 is constructed by points at the triangle face centers of the 122.

These polytopes are from a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

## 1_22 polytope

The 1_22 polytope contains 72 vertices, and 54 5-demicubic facets. It has a birectified 5-simplex vertex figure. Its 72 vertices represent the root vectors of the simple Lie group E6.

## Alternate names

• Pentacontatetra-peton (Acronym Mo) - 54-facetted polypeton (Jonathan Bowers)
• ## Construction

It is created by a Wythoff construction upon a set of 6 hyperplane mirrors in 6-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram, .

Removing the node on either of 2-length branches leaves the 5-demicube, 131, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, 022, .

## Related complex polyhedron

The regular complex polyhedron 3{3}3{4}2, , in C 2 has a real representation as the 122 polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its complex reflection group is 3[3]3[4]2, order 1296. It has a half-symmetry quasiregular construction as , as a rectification of the Hessian polyhedron, .

## Related polytopes and honeycomb

Along with the semiregular polytope, 221, it is also one of a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

## Geometric folding

The 122 is related to the 24-cell by a geometric folding E6 → F4 of Coxeter-Dynkin diagrams, E6 corresponding to 122 in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 122.

## Tessellations

This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, 222, .

## Rectified 1_22 polytope

The rectified 122 polytope (also called 0221) can tessellate 6-dimensional space as the Voronoi cell of the E6* honeycomb lattice (dual of E6 lattice).

## Alternate names

• Birectified 221 polytope
• Rectified pentacontatetrapeton (acronym Ram) - rectified 54-facetted polypeton (Jonathan Bowers)
• ## Construction

Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .

Removing the ring on the short branch leaves the birectified 5-simplex, .

Removing the ring on the either 2-length branch leaves the birectified 5-orthoplex in its alternated form: t2(211), .

The vertex figure is determined by removing the ringed node and ringing the neighboring ring. This makes 3-3 duoprism prism, {3}×{3}×{}, .

## Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

## Alternate names

• Truncated 122 polytope
• ## Construction

Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .

## Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

## Alternate names

• Bicantellated 221
• Birectified pentacontitetrapeton (barm) (Jonathan Bowers)
• ## Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

## Alternate names

• Tricantellated 221
• Trirectified pentacontitetrapeton (trim) (Jonathan Bowers)
• ## References

1 22 polytope Wikipedia

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